Identify the States
The states of the Markov chain represent the machine that uses only digits \(0\) and \(1\), and these digits can either change or stay the same at each stage.
State 0: The current digit is \(0\).
State 1: The current digit is \(1\).
Define Transition Probabilities
The probabilities of transitioning from one state to another:
Probability \(p\): The digit changes at any stage.
Probability \(q = 1 - p\): The digit remains the same at any stage.
The transitions between states:
From \(0\) to \(0\) (Stays the Same): This happens with probability \(q\), as the digit does not change.
From \(0\) to \(1\) (Changes): This occurs with probability \(p\), as the digit changes from \(0\) to \(1\).
From \(1\) to \(1\) (Stays the Same): Similar to the first case, this has a probability \(q\).
From \(1\) to \(0\) (Changes): Also occurs with probability \(p\), as the digit changes from \(1\) to \(0\).
Form the Transition Matrix
The transition matrix \(P\) is formed by placing the transition probabilities in a matrix corresponding to transitions from one state to another. Let’s label the states such that \(0\) is the first state and \(1\) is the second state:
\[ P = \begin{pmatrix} q & p \\ p & q \end{pmatrix} \]
Given \(q = 1 - p\), we can also express the matrix in terms of \(p\) alone:
\[ P = \begin{pmatrix} 1-p & p \\ p & 1-p \end{pmatrix} \]
Interpret the Transition Matrix
The first row corresponds to starting in state \(0\). It shows the probability of staying in state \(0\) as \(1-p\) and the probability of transitioning to state \(1\) as \(p\).
The second row corresponds to starting in state \(1\). It shows the probability of transitioning to state \(0\) as \(p\) and the probability of staying in state \(1\) as \(1-p\).